Transcript Document

2010 VDOE Mathematics
Institute
Grades 6-8
Focus: Patterns, Functions, and Algebra
Fall 2010
Content Focus
Key changes at the middle school level:
• Properties of Operations with Real Numbers
• Equations and Expressions
• Inequalities
• Modeling Multiplication and Division of
Fractions
• Understanding Mean: Fair Share and
Balance Point
• Modeling Operations with Integers
Fall 2010
2
Supporting Implementation of
2009 Standards
• Highlight key curriculum changes.
• Connect the mathematics across grade levels.
• Model instructional strategies.
Fall 2010
3
Properties of Operations
Fall 2010
4
Properties of Operations: 2001 Standards
7.3 The student will identify and apply the following properties of
operations with real numbers:
a) the commutative and associative properties for addition and
multiplication;
3.20a&b; 4.16b
b) the distributive property;
5.19
c) the additive and multiplicative identity properties;
6.19a
d) the additive and multiplicative inverse properties; and
e) the multiplicative property of zero.
6.19c
6.19b
8.1 The student will
a) simplify numerical expressions involving positive exponents,
using rational numbers, order of operations, and properties of
operations with real numbers;
Fall 2010
5
Properties of Operations: 2009 Standards
3.20
b) Identify examples of the identity and commutative properties for addition and
multiplication.
4.16b b) Investigate and describe the associative property for addition and multiplication.
5.19
6.19
7.16
8.1a
Investigate and recognize the distributive property of multiplication over addition.
Investigate and recognize
a) the identity properties for addition and multiplication;
b) the multiplicative property of zero; and
c) the inverse property for multiplication.
Apply the following properties of operations with real numbers:
a) the commutative and associative properties for addition and multiplication;
b) the distributive property;
c) the additive and multiplicative identity properties;
d) the additive and multiplicative inverse properties; and
e) the multiplicative property of zero.
a) simplify numerical expressions involving positive exponents, using rational numbers,
order of operations, and properties of operations with real numbers;
8.15c c) identify properties of operations used to solve an equation.
Fall 2010
6
3.20a&b: Identity Property for Multiplication
Fall 2010
x,÷
1
2
3
4
5
6
7
8
9
10
11
12
1
1
2
3
4
5
6
7
8
9
10
11
12
2
2
4
6
8
10
12
14
16
18
20
22
24
3
3
6
9
12
15
18
21
24
27
30
33
36
4
4
8
12
16
20
24
28
32
36
40
44
48
5
5
10
15
20
25
30
35
40
45
50
55
60
6
6
12
18
24
30
36
42
48
54
60
66
72
7
7
14
21
28
35
42
49
56
63
70
77
84
8
8
16
24
32
40
48
56
9
9
18
27
36
10
10
20
30
40
11
11
22
33
44
12
12
24
36
48
60
72
84
64 72 80 88 96
The
first row and
45
54 63 of72products
81 90 99in108
column
a
chart
50 60 multiplication
70 80 90 100 110
120
illustrate the identity
55 66 77 88 99 110 121 132
property.
96 108 120 132 144
7
3.20a&b: Commutative Property for Multiplication
x,÷
1
2
3
4
5
6
7
8
9
10
11
12
1
1
2
3
4
5
6
7
8
9
10
11
12
4
6
8
10
12
14
16
18
20
22
24
9
12
15
18
21
24
27
30
33
36
16
20
24
28
32
36
40
44
48
25
30
35
40
45
50
55
60
36
42
48
54
60
66
72
49
56
63
70
77
84
64
72
80
88
96
81
90
99
108
100
110
120
121
132
2
3
4
5
6
7
8
Why does the
9
diagonal
of perfect
10
squares
form a line of
11
symmetry
in the
chart?
12
Fall 2010
144
8
3.20a&b: Commutative Property for Multiplication
Fall 2010
x,÷
1
2
3
4
5
6
7
8
9
10
11
12
1
1
2
3
4
5
6
7
8
9
10
11
12
2
2
4
6
8
10
12
14
16
18
20
22
24
3
3
6
9
12
15
18
21
24
27
30
33
36
4
4
8
12
16
20
24
28
32
36
40
44
48
5
5
10
15
20
25
30
35
40
45
50
55
60
6
6
12
18
24
30
36
42
48
54
60
66
72
7
7
14
21
28
35
42
49
56
63
70
77
84
8
8
16
24
32
40
48
9
9
18
27
36
45
10
10
20
30
40
50
11
11
22
33
44
55
12
12
24
36
48
60
64 red
72 rectangle
80
88
96
The
54 (4x6)
63
72 and
81
90the99blue
108
rectangle
(6x4)110both
60
70
80
90 100
120
cover
an
area
of 24
66
77
88
99
110
121 132
squares on the
72
84
96 108 120 132 144
multiplication chart.
56
9
6.19: Multiplicative Property of Zero
x,÷
1
2
3
4
5
6
7
8
9
10
11
12
1
1
2
3
4
5
6
7
8
9
10
11
12
2
2
4
6
8
10 12 14 16
18
20
22
24
3
3
6
9
12 15 18 21 24
27
30
33
36
4
4
8
12 16 20 24 28 32
36
40
44
48
5
5
10 15 20 25 30 35 40
45
50
55
60
6
6
12 18 24 30 36 42 48
54
60
66
72
7
7
14 21 28 35 42 49 56
63
70
77
84
8
8
16 24 32 40 48 56 64
72
80
88
96
9
9
18 27 36 45 54 63 72
10 10
11
11
12 12
Fall 2010
Area multiplication
is based on
81 90 99 108
rectangles. If one factor is
20 30 40 50 60 70 80 90 100 110 120
zero, then the number sentence
22 33 44 55 66 77 88 99 110 121 132
doesn’t describe a rectangle, it
24 36 48 60 72 84 96 108 120 132 144
describes a line segment, and
the product (the “area”) is zero.
10
Meanings of Multiplication
For 5 x 4 = 20…
Repeated Addition: “4, 8, 12, 16, 20.”
Groups-Of: “Five bags of candy with four pieces of candy in
each bag.”
Rectangular Array: “Five rows of desks with four desks in each
row.”
Rate: “Dave bought five raffle tickets at $4.00 apiece.” or “Dave
walked four miles per hour for five hours.”
Comparison: “Alice has 4 cookies; Ralph has five times as
many.”
Combinations: “Cindy has five different shirts and four different
pairs of pants; how many different shirt/pants outfits can she
make?”
Area: “Ricky buys a rectangular rug 5 feet long and 4 feet wide.”
Adapted from Baroody, Arthur J., Fostering Children’s Mathematical Power,
LEA Publishing, 1998, Chapter 5.
Fall 2010
11
3.6: Represent Multiplication Using an Area Model
Use your base ten blocks to represent
3 x 6 = 18
National Library of Virtual Manipulatives – Rectangle Multiplication
Fall 2010
12
3.6: Represent Multiplication Using an Area Model
Or did yours look like this?
Rotating the rectangle doesn’t change its area.
Commutative
Property:
National Library of Virtual Manipulatives – Rectangle Multiplication
Fall 2010
13
3.6: Represent Multiplication Using an Area Model
Use your base ten blocks to represent
5 x 14 = 70
What is the area
of the red inner
rectangle?
What is the area
of the blue inner
rectangle?
National Library of Virtual Manipulatives – Rectangle Multiplication
Fall 2010
14
3.6:5.19:
Represent
Distributive
Multiplication
Property Using
of Multiplication
an Area Model
How could students record the area of
the 5 x 14 rectangle?
5 x 4 = 20
5 x 10 = 50
Fall 2010
14
x5
5 x 10 → 50
5 x 4 → + 20
70
15
5.19: Distributive Property
of Multiplication Over Addition
Understanding the Standard: “The distributive property
states that multiplying a sum by a number gives the same
result as multiplying each addend by the number and then
adding the products (e.g., 3(4 + 5) = 3 x 4 + 3 x 5, 5 x (3 + 7)
= (5 x 3) + (5 x 7); or (2 x 3) + (2 x 5) = 2 x (3 + 5).”
Essential Knowledge & Skills:
• “Investigate and recognize the distributive property of whole
numbers, limited to multiplication over addition, using
diagrams and manipulatives.”
• “Investigate and recognize an equation that represents the
distributive property, when given several whole number
equations, limited to multiplication over addition.”
National Library of Virtual Manipulatives – Rectangle Multiplication
Fall 2010
16
5.19: Distributive Property
of Multiplication Over Addition
Use base ten blocks to build a
12 x 23 rectangle.
The traditional multidigit multiplication
algorithm finds the sum
of the areas of two inner
rectangles.
National Library of Virtual Manipulatives – Rectangle Multiplication
Fall 2010
17
5.19: Distributive Property
of Multiplication Over Addition
The partial products algorithm finds the sum of the areas
of four inner rectangles.
Look familiar?
F.irst
O.uter
I.nner
L.ast
National Library of Virtual Manipulatives – Rectangle Multiplication
Fall 2010
18
Strengths of the Area Model of Multiplication
Illustrates the inherent connections between
multiplication and division:
• Factors, divisors, and quotients are represented by the
lengths of the rectangle’s sides.
• Products and dividends are represented by the area of
the rectangle.
Versatile:
• Can be used with whole numbers and decimals (through
hundredths).
• Rotating the rectangle illustrates commutative property.
• Forms the basis for future modeling: distributive
property; factoring with Algebra Tiles; and Completing
the Square to solve quadratic equations.
Fall 2010
19
4.16b: Associative Property for Multiplication
Use your base ten blocks to build a rectangular solid
2cm by 3cm by 4cm
Base: 3cm by 4cm; Height: 2cm
Volume: 2 x (3 x 4) = 24 cm3
Associative Property: The
Base: 2cm by 3cm; Height: 4cm grouping of the factors does
Volume: (2 x 3) x 4 = 24 cm3
not affect the product.
National Library of Virtual Manipulatives – Space Blocks
Fall 2010
20
Expressions and Equations
Fall 2010
A Look At Expressions and Equations
A manipulative, like algebra tiles,
creates a concrete foundation for
the abstract, symbolic
representations students begin to
wrestle with in middle school.
22
Fall 2010
What do these tiles represent?
1 unit
Area = 1 square unit
1 unit
Tile Bin
Unknown length, x units
Area = x square units
1 unit
x units
x units
Area = x2 square units
The red tiles denote negative quantities.
Fall 2010
23
Modeling expressions
x+5
Tile Bin
5+x
Fall 2010
24
Modeling expressions
x-1
Fall 2010
Tile Bin
25
Modeling expressions
x+2
Tile Bin
2x
Fall 2010
26
Modeling expressions
x2 + 3x + 2
Fall 2010
Tile Bin
27
Simplifying expressions
x2 + x - 2x2 + 2x - 1
Tile Bin
zero pair
Simplified expression
-x2 + 3x - 1
Fall 2010
28
Simplifying expressions
2(2x + 3)
Tile Bin
Simplified expression
4x + 6
Fall 2010
29
Two methods of illustrating the
Distributive Property:
Example: 2(2x + 3)
Fall 2010
30
Solving Equations
How does this concept progress as we move through middle school?
6th grade:
6.18 The student will solve one-step linear equations in one variable involving whole
number coefficients and positive rational solutions.
7th grade:
7.14 The student will
a)
solve one- and two-step linear equations in one variable; and
b)
solve practical problems requiring the solution of one- and two-step linear
equations.
8th grade:
8.15 The student will
a) solve multistep linear equations in one variable on one and two sides of the
equation;
b) solve two-step linear inequalities and graph the results on a number line; and
c) identify properties of operations used to solve an equation.
Fall 2010
31
Solving Equations
Tile Bin
Fall 2010
32
Solving Equations
6.18 The student will solve one-step linear equations in one variable involving whole
number coefficients and positive rational solutions.
x+3=5
Fall 2010
Tile Bin
33
Solving Equations
6.18 The student will solve one-step linear equations in one variable involving whole
number coefficients and positive rational solutions.
Pictorial Representation:
Symbolic Representation:
Condensed Symbolic Representation:
x+3=5
x+3=5
̵3 ̵3
x+3=5
̵3 ̵3
x=2
x=2
Fall 2010
34
Solving Equations
6.18 The student will solve one-step linear equations in one variable involving whole
number coefficients and positive rational solutions.
2x = 8
Fall 2010
Tile Bin
35
Solving Equations
7.14
The student will solve one- and two-step linear equations in one variable;
and solve practical problems requiring the solution of one- and two-step linear equations.
3=x-1
Fall 2010
Tile Bin
36
Solving Equations
7.14
The student will solve one- and two-step linear equations in one variable;
and solve practical problems requiring the solution of one- and two-step linear equations.
2x + 3 = 13
Fall 2010
Tile Bin
37
Solving Equations
7.14
The student will solve one- and two-step linear equations in one variable;
and solve practical problems requiring the solution of one- and two-step linear equations.
Pictorial Representation:
Symbolic Representation:
Condensed Symbolic Representation:
2x + 3 = 13
2x + 3 = 13
̵3 ̵3
2x = 10
2
2
2x + 3 = 13
̵3 ̵3
2x = 10
2
2
x=5
x=5
Fall 2010
38
Solving Equations
7.14
The student will solve one- and two-step linear equations in one variable;
and solve practical problems requiring the solution of one- and two-step linear equations.
0 = 4 – 2x
Fall 2010
Tile Bin
39
Solving Equations
7.14
The student will solve one- and two-step linear equations in one variable;
and solve practical problems requiring the solution of one- and two-step linear equations.
Pictorial Representation:
Symbolic Representation:
Condensed Symbolic Representation:
0 = 4 – 2x
0 = 4 – 2x
̵4 ̵4
-4 = -2x
2
2
0 = 4 – 2x
̵4 ̵4
-4 = -2x
-2 -2
2=x
-2 = -x
2=x
Fall 2010
40
Solving Equations
8.15 The student will
a) solve multistep linear equations in one variable on one and two sides of the equation;
b) solve two-step linear inequalities and graph the results on a number line; and
c) identify properties of operations used to solve an equation.
3x + 5 – x = 11
Fall 2010
Tile Bin
41
Solving Equations
8.15 The student will
a) solve multistep linear equations in one variable on one and two sides of the equation;
b) solve two-step linear inequalities and graph the results on a number line; and
c) identify properties of operations used to solve an equation.
Pictorial Representation:
Symbolic Representation:
Condensed Symbolic Representation:
3x + 5 – x = 11
2x + 5 = 11
2x + 5 = 11
-5 -5
2x = 6
2
2
3x + 5 – x = 11
2x + 5 = 11
-5 -5
2x = 6
2
2
x=3
x=3
Fall 2010
42
Solving Equations
8.15 The student will
a) solve multistep linear equations in one variable on one and two sides of the equation;
b) solve two-step linear inequalities and graph the results on a number line; and
c) identify properties of operations used to solve an equation.
x + 2 = 2(2x + 1)
Fall 2010
Tile Bin
43
Solving Equations
8.15 The student will
a) solve multistep linear equations in one variable on one and two sides of the equation;
b) solve two-step linear inequalities and graph the results on a number line; and
c) identify properties of operations used to solve an equation.
Pictorial Representation:
Symbolic Representation:
x + 2 = 2(2x + 1)
x + 2 = 4x + 2
x + 2 = 4x + 2
-x
-x
x + 2 = 2(2x + 1)
x + 2 = 4x + 2
-x
-x
2 = 3x + 2
-2
-2
2 = 3x + 2
-2
-2
0 = 3x
3 3
0 = 3x
3 3
0=x
Fall 2010
Condensed Symbolic Representation:
0=x
44
Modeling Multiplication and
Division of Fractions
Fall 2010
45
So what’s new about fractions
in Grades 6-8?
SOL 6.4
The student will demonstrate multiple
representations of multiplication and
division of fractions.
Fall 2010
46
Thinking About Multiplication
The
expression…
We read it…
It means…
It looks like…
23
1
2
3
1 1

2 3
Fall 2010
47
Thinking About Multiplication
The
expression…
23
1
2
3
1 1

2 3
Fall 2010
We read it…
It means…
2 times 3
two groups of
three
2 times
1
3
1 times 1
2
3
It looks like…
two groups of
one-third
one-half group
of one-third
48
Making sense of multiplication
of fractions using paper
folding and area models
Enhanced Scope and Sequence,
2004, pages 22 - 24
Fall 2010
49
Making sense of multiplication
of fractions using paper
folding and area models
Enhanced Scope and Sequence,
2004, pages 22 - 24
Fall 2010
50
Making sense of multiplication
of fractions using paper
folding and area models
Enhanced Scope and Sequence,
2004, pages 22 - 24
Fall 2010
51
The Importance of Context
• Builds meaning for operations
• Develops understanding of and
helps illustrate the relationships
among operations
• Allows for a variety of approaches
to solving a problem
Fall 2010
52
Contexts for Modeling
Multiplication of Fractions
The Andersons had pizza for dinner,
and there was one-half of a pizza left
over. Their three boys each ate
one-third of the leftovers for a late
night snack.
How much of the original pizza did
each boy get for snack?
Fall 2010
53
1 1
1
 
3 2
6
One-third of one-half of a pizza is equal to one-sixth of a pizza.
Which
meaning of
multiplication
does this
model fit?
Fall 2010
54
Another Context for
Multiplication of Fractions
Andrea and Allison are partners in a relay
race. Each girl will run half the total
distance. On race day, Andrea stops for
water after running 1 of her half of the
3
race.
What portion of the race had Andrea run
when she stopped for water?
Fall 2010
55
1 1
1
 
3 2
6
Students need experiences with problems
that lend themselves to a linear model.
Fall 2010
56
Another Context for
Multiplication of Fractions
Mrs. Jones has 24 gold stickers that
she bought to put on perfect test
1
papers. She took 2 of the stickers
out of the package, and then she
1
used 3 of that half on the papers.
What fraction of the 24 stickers did
she use on the perfect test papers?
Fall 2010
57
1 1
1
 
3 2
6
One-third of one-half of the 24 stickers is
1
6
of the 24 stickers.
What
meaning(s) of
multiplication
does this
model fit?
Problems involving discrete items
may be represented with set models.
Fall 2010
58
What’s the relationship between
multiplying and dividing?
Multiplication and division are inverse
relations
One operation undoes the other
Division by a number yields the same
result as multiplication by its reciprocal
(inverse). For example:
1
62  6
2
Fall 2010
59
Meanings of Division
For 20 ÷ 5 = 4…
Divvy Up (Partitive): “Sally has 20 cookies. How many
cookies can she give to each of her five friends, if she gives
each friend the same number of cookies?
- Known number of groups, unknown group size
Measure Out (Quotitive): “Sally has 20 minutes left on
her cell phone plan this month. How many more 5-minute calls
can she make this month?
- Known group size, unknown number of groups
Adapted from Baroody, Arthur J., Fostering Children’s Mathematical Power,
LEA Publishing, 1998.
Fall 2010
60
Sometimes, Always, Never?
• When we multiply, the product is
larger than the number we start with.
• When we divide, the quotient is
smaller than the number we start with.
Fall 2010
61
“I thought times makes it bigger...”
When moving beyond whole numbers to situations involving
fractions and mixed numbers as factors, divisors, and
dividends, students can easily become confused. Helping
them match problems to everyday situations can help them
better understand what it means to multiply and divide with
fractions. However, repeated addition and array meanings of
multiplication, as well as a divvy up meaning of division, no
longer make as much sense as they did when describing
whole number operations.
Using a Groups-Of interpretation of multiplication and a
Measure Out interpretation of division can help:
Adapted from Baroody, Arthur J., Fostering Children’s Mathematical Power,
LEA Publishing, 1998.
Fall 2010
62
“Groups of” and “Measure Out”
1/4 x 8: “I have one-fourth of a box of 8 doughnuts.”
8 x 1/4: “There are eight quarts of soda on the table. How many whole gallons of soda are
there?”
1/2 x 1/3: “The gas tank on my scooter holds 1/3 of a gallon of gas. If I have 1/2 a tank left, what
fraction of a gallon of gas do I have in my tank?”
1¼ x 4: “Red Bull comes in packs of four cans. If I have 1¼ packs of Red Bull, how many cans
do I have?”
3½ x 2½: “If a cross country race course is 2½ miles long, how many miles have I run after 3½
laps?
3/4 ÷ 2: “How much of a 2-hour movie can you watch in 3/4 of an hour?” *This type may be
easier to describe using divvy up.
2 ÷ 3/4: “How many 3/4-of-an-hour videos can you watch in 2 hours?”
3/4 ÷ 1/8: “How many 1/8-sized (of the original pie) pieces of pie can you serve from 3/4 of a
pie?”
2½ ÷ 1/3: “A brownie recipe calls for 1/3 of a cup of oil per batch. How many batches can you
make if you have 2½ cups of oil left?”
Fall 2010
63
Thinking About Division
The
expression…
We read it…
It means…
It looks like…
20 ÷ 5
20 
1
2
Fall 2010
64
Thinking About Division
The
expression…
20 ÷ 5
We read it…
20 divided
by 5
It means…
It looks like…
20 divided into
groups of 5;
20 divided into 5
equal groups…
How many 5’s are
in 20?
1
20 
2
20 divided
by 1
2
20 divided into
groups of 1 …
2
How many 1 ’s are
2
in 20?
65
Fall 2010
65
Thinking About Division
The expression…
1 1

2 3
We read it…
one-half
divided by
one-third
It means…
It looks like…
1
2
divided into
groups of 1 …
3
?
How many 1 ’s are
3
1
in 2 ?
Is the quotient more than one
or less than one? How do you
know?
Fall 2010
66
Contexts for
Division of Fractions
The Andersons had half of a pizza left after
1
dinner. Their son’s typical serving size is 3
pizza. How many of these servings will he eat
if he finishes the pizza?
Fall 2010
67
1 1
1
 1
2 3
2
1
1
1
2 pizza divided into 3 pizza servings = 1 2 servings
1 serving
1
2 serving
Fall 2010
68
Another Context for
Division of Fractions
Marcy is baking brownies. Her recipe calls for
1
3 cup cocoa for each batch of brownies.
Once she gets started, Marcy realizes she
1
only has 2 cup cocoa. If Marcy uses all of the
cocoa, how many batches of brownies can
she bake?
Fall 2010
69
1 1
1
 1
2 3
2
1 cup
Three batches (or
Two batches (or
1
2
cup
3
3
cup)
2
3
cup)
1
1
2
batches
One batch (or 13 cup)
0 cups
Fall 2010
70
Another Context for
Division of Fractions
1
2
Mrs. Smith had of a sheet cake left over after
her party. She decides to divide the rest of the
1
cake into portions that equal 3 of the original
cake.
1
3
How many cake portions can Mrs. Smith make
from her left-over cake?
Fall 2010
71
What could it look like?
1 1

2 3
Fall 2010
72
What does it look
like numerically?
Fall 2010
73
What is the role of common
denominators in dividing
fractions?
Ensures division of the same size
units
Assist with the description of parts
of the whole
Fall 2010
74
What about
the traditional algorithm?
•If the traditional “invert and multiply” algorithm is
taught, it is important that students have the
opportunity to consider why it works.
•Representations of a pictorial nature provide a
visual for finding the reciprocal amount in a given
situation.
•The common denominator method is a different,
valid algorithm. Again, it is important that students
have the opportunity to consider why it works.
Fall 2010
75
What about
the traditional algorithm?
Build understanding:
1
Think about 20 ÷ 2 .
How many one-half’s are in 20?
How many one-half’s are in each of the 20 individual wholes?
Experiences with fraction divisors having a
numerator of one illustrate the fact that within each
unit, the divisor can be taken out the reciprocal
number of times.
Fall 2010
76
What about
the traditional algorithm?
Later, think about divisors with numerators > 1.
Think about 1 ÷
2
.
3
2
How many times could we take 3 from 1?
1
We can take it out once, and we’d have 3 left. We
2
3
could only take half of another
from the remaining
3
portion. That’s a total of 2 .
3
2
In each unit, there are sets of .
2
3
Fall 2010
77
Multiple Representations
Instructional programs from pre-k through grade
12 should enable all students to –
•Create and use representations to organize,
record and communicate mathematical ideas;
•Select, apply, and translate among
mathematical representations to solve problems;
•Use representations to model and interpret
physical, social, and mathematical phenomena.
from Principles and Standards for School Mathematics (NCTM, 2000), p. 67.
Fall 2010
78
Using multiple representations
to express understanding
Given problem
Contextual situation
Check your solution
Solve numerically
Fall 2010
Solve graphically
79
Using multiple
representations
to express
understanding
of division of
fractions
Fall 2010
80
Mean:
Fair Share and Balance Point
Fall 2010
81
Mean: Fair Share
2009 5.16: The student will
a) describe mean, median, and mode as measures
of center;
b) describe mean as fair share;
c) find the mean, median, mode, and range of a set
of data; and
d) describe the range of a set of data as a measure
of variation.
Understanding the Standard: “Mean represents a fair
share concept of the data. Dividing the data constitutes a fair
share. This is done by equally dividing the data points. This
should be demonstrated visually and with manipulatives.”
Fall 2010
82
Understanding the Mean
Each person at the table should:
1. Grab one handful of snap cubes.
2. Count them and write the number
on a sticky note.
3. Snap the cubes together to form a
train.
Fall 2010
83
Understanding the Mean
Work together at your table to answer
the following question:
If you redistributed all of the cubes
from your handfuls so that everyone
had the same amount (so that they
were “shared fairly”), how many
cubes would each person receive?
Fall 2010
84
Understanding the Mean
What was your answer?
- How did you handle “leftovers”?
- Add up all of the numbers from the
original handfuls and divide the sum
by the number of people at the table.
- Did you get the same result?
- What does your answer represent?
Fall 2010
85
Understanding the Mean
Take your sticky note and place it on
the wall, so they are ordered…
Horizontally: Low to high, left to
right; leave one space if there is a
missing number.
Vertically: If your number is
already on the wall, place your
sticky note in the next open
space above that number.
Fall 2010
86
Understanding the Mean
How did we display our data?
2009 3.17c
Fall 2010
87
Understanding the Mean
Looking at our line plot, how can we
describe our data set? How can we
use our line plot to:
- Find the range?
- Find the mode?
- Find the median?
- Find the mean?
Fall 2010
88
Mean: Balance Point
2009 6.15: The student will
a) describe mean as balance point; and
b) decide which measure of center is appropriate
for a given purpose.
Understanding the Standard: “Mean can be defined as the
point on a number line where the data distribution is balanced.
This means that the sum of the distances from the mean of all
the points above the mean is equal to the sum of the
distances of all the data points below the mean.”
Essential Knowledge & Skills:
• Identify and draw a number line that demonstrates the
concept of mean as balance point for a set of data.
Fall 2010
89
Where is the balance point for this
data set?
X
Fall 2010
X
X
X
X X
90
Where is the balance point for this
data set?
X
X
X
X
Fall 2010
X
X
91
Where is the balance point for this
data set?
X
X
X
Fall 2010
X
X X
92
Where is the balance point for this
data set?
X
X
Fall 2010
X
X
X
X
93
Where is the balance point for this
data set?
3 is the
Balance Point
X
Fall 2010
X
X
X
X
X
94
Where is the balance point for this
data set?
X
Fall 2010
X
X
X
X X
95
Where is the balance point for this
data set?
Move 2 Steps
Move 2 Steps
Move 2 Steps
Move 2 Steps
Fall 2010
4 is the Balance Point
96
We can confirm this by calculating:
2 + 2 + 2 + 3 + 3 + 4 + 5 + 7 + 8 = 36
36 ÷ 9 = 4
Fall 2010
The Mean is the Balance Point
97
Where is the balance point for this
data set?
If we could “zoom in” on the
Move 1 Step
The Balance Point is
between 10 and 11
Move 2 Steps
(closer to 10).
Fall 2010
space between 10 and 11, we
could continue this process to
arrive at a decimal value for the
balance point.
Move 2 Steps
Move 1 Step
98
Mean: Balance Point
When demonstrating finding the balance point:
1. CHOOSE YOUR DEMONSTRATION DATA SETS
INTENTIONALLY.
2. Use a line plot to represent the data set.
3. Begin with the extreme data points.
4. Balance the moves, moving one data point from each
side an equal number of steps toward the center.
5. Continue until the data is distributed symmetrically or until
there are only two values left on the line plot.
Fall 2010
99
Assessing Higher-Level
Thinking
Key Points for 2009 5.16 & 6.15:
Students still need to be able to calculate the mean by
summing up and dividing, but they also need to
understand:
- why it’s calculated this way (“fair share”);
- how the mean compares to the median and the
mode for describing the center of a data set; and
- when each measure of center might be used
to represent a data set.
Fall 2010
10
Mean:
Fair Share & Balance Point
“Students need to understand that the mean ‘evens out’ or
‘balances’ a set of data and that the median identifies the
‘middle’ of a data set. They should compare the utility of the
mean and the median as measures of center for different
data sets. …students often fail to apprehend many subtle
aspects of the mean as a measure of center. Thus, the
teacher has an important role in providing experiences that
help students construct a solid understanding of the mean
and its relation to other measures of center.”
- NCTM Principles & Standards for School Mathematics, p. 250
Fall 2010
10
Inequalities
Fall 2010
102
Inequalities
SOL 6.20
The student will graph inequalities on a number line.
SOL 7.15
The student will
a) solve one-step inequalities in one variable; and
b) graph solutions to inequalities on the number line.
SOL 8.15
The student will
b) solve two-step linear inequalities and graph the
results on a number line
Fall 2010
103
Inequalities
What does inequality mean in the
world of mathematics?
mathematical sentence comparing
two unequal expressions
How are they used in everyday life?
to solve a problem or describe a
relationship for which there is
more than one solution
Fall 2010
104
Equations vs. Inequalities
x=2
x>2
How are they alike?
How are they different?
So, what about x > 2?
Fall 2010
105
Equations vs. Inequalities
x=2
x>2
x>2
Fall 2010
106
Open or Closed?
x > 16
-5 > y
m > 12
n < 341
-3 < j
and, which way should the ray go?
Fall 2010
107
Equations vs. Inequalities
x+2=8 x+2<8
How are they alike?
How are they different?
So, what about x + 2 < 8?
Fall 2010
108
Equations vs. Inequalities
x+2=8
x+2<8
How are they alike?
Both statements include the terms: x, 2 and 8
The solution set for both statements involves 6.
How are they different?
The solution set for x + 2 = 8 only includes 6. The solution
set for x + 2 < 8 does includes all real numbers less than 6.
What about x + 2 < 8?
The solution set for this inequality includes 6 and
all real numbers less than 6.
Fall 2010
109
Equations vs. Inequalities
x+ 2 = 8
x+ 2 < 8
x+ 2 < 8
Fall 2010
110
Inequality Match
With your tablemates, find as
many matches as possible in
the set of cards.
Fall 2010
111
X >5
X is greater
than 5
SAMPLE MATCH
Fall 2010
112
Operations with Integers
Fall 2010
113
Operations with Integers
2009 7.3a: The student will
a) model addition, subtraction, multiplication and
division of integers; and
b) add, subtract, multiply, and divide integers.
Is this really a “new” SOL?
2001 7.5: The student will formulate rules for and solve
practical problems involving basic operations (addition,
subtraction, multiplication, and division) with integers.
“Model”
Fall 2010
114
Assessing Higher-Level
Thinking
7.3a: The student will model addition, subtraction,
multiplication, and division of integers.
= -1
=1
What operation does this model?
Fall 2010
3 + (-7) = -4
115
Assessing Higher-Level
Thinking
7.3a: The student will model addition, subtraction,
multiplication, and division of integers.
=1
= -1
3 • (-4) = -12
What operation does this model?
Fall 2010
116
Assessing Higher-Level
Thinking
7.3a: The student will model addition, subtraction,
multiplication, and division of integers.
5 5+ -(-17)
17 = =-12
-12
What operation does this model?
Fall 2010
117
Assessing Higher-Level
Thinking
7.3a: The student will model addition, subtraction,
multiplication, and division of integers.
3 • (-5) = -15
What operation does this model?
Fall 2010
118
Another Example of Assessing
Higher-Level Thinking
7.5c: The student will describe how changing one
measured attribute of a rectangular prism affects its
volume and surface area.
Describe how the volume of
the rectangular prism
shown (height = 8 in.) would
be affected if the height was
increased by a scale factor
of ½ or 2.
8 in.
3 in.
5 in.
Fall 2010
119
Tying it All Together
1. Improved vertical alignment of
content with increased cognitive
demand.
2. Key conceptual models can be
extended across grade levels.
3. Refer to the Curriculum Framework.
4. Pay attention to the changes in the
verbs.
Fall 2010
120
Exit Slip
1. Aha...
2. Can’t wait to share…
3. HELP!
Fall 2010
121